Size functor: A homological shape descriptor
by
Francesca Cagliari
Dipartimento di Matematica, Universitá di Bologna, Italy
Coauthors: Massimo Ferri

The talk introduces a functor as a new shape descriptor. This is a categorical evolution of size functions, descriptors already applied in several pattern recognition areas . The leading ideas are:

• ``shape'' is not just an object, but also the behaviour of some (continuous real) functions on it;
• shape comparison is best performed if the ``story'' of the functions - at the various levels - is condensed in a formal, comparable construction.

For size functions, the construction reflected the ranks of the images of the H₀-homology morphisms induced by inclusion . Here a much wider setting is proposed, through extensive use of Morse theory.

Let Morse denote the subcategory of Diff/R whose objects are pairs (M, f) where M is a compact, connected manifold, and f is a Morse function on M, injective on the set of critical points, and whose morphisms are maps g between manifolds, commuting with the respecting Morse functions. Let ES be the category of exact sequences of abelian groups.

We define a functor S:Morse --> Funct(Rord, Exact).

Given a Morse function f on a manifold M, for each x in R let x₀, x₁ in R be the two critical values of f such that x₀ < x₁ <= x, and there are no other critical values in the interval [x₀, x], but x₀, x₁.

It is well-known that both M_x and M_x1 have the homotopy type of the attachment space of M_x0 with a \lambda-cell, where \lambda is the index of the critical point.

The functor F=S((M, f)) is defined to associate to each x (values x below the second lowest critical value are treated aside) the Mayer-Vietoris sequence corresponding to the attachment space just described. On a morphism g its value is the induced homology morphism H(g).

The functor S portraits the development of the manifold through its critical points; moreover, it keeps explicitly into account the topology ofthe various M_x. This is what we consider to be essential for studying the ``shape'' of (M, f).

The presentation of each H_k(M_x) can be reconstructed through the sequence of critical values, by applying the Mayer-Vietoris theorem to each cell attachment. Each critical point contributes with either a generator or a relator .

We envisage, as a development, the definition of distances between images S((M, f)) in order to allow classifications, queries and other Pattern Recognition tasks.

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P. Frosini, Connections between size functions and critical points, Math. Meth. Appl. Sci., 19 (1996), 555-569.

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F. Cagliari, C. Landi, M. Grasselli, Presentation of Morse homology for studying shape of manifold preprint

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F. Cagliari, M. Ferri, P. Pozzi, Size functions from a categorical viewpoint Acta Appl. Math.

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M. Schwarz, ``Morse Homology'', Progress in mathematics; Vol III, Birkhäuser, 1993.

Date received: February 28, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-46.